JEE Main & Advanced Mathematics Vector Algebra Question Bank

done Vector triple product Total Questions - 24

• question_answer1)
  \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is coplanar with
  A)
           b and c done clear
  B)
           c and a done clear
  C)
           a and b done clear
  D)
           a, b and c done clear
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• question_answer2)
  If  \[\mathbf{u}=\mathbf{i}\times (\mathbf{a}\times \mathbf{i})+\mathbf{j}\times (\mathbf{a}\times \mathbf{j})+\mathbf{k}\times (\mathbf{a}\times \mathbf{k}),\]then [RPET 1989, 97; MNR 1986, 93; MP PET 1987, 98, 99, 2004; UPSEAT 2000, 02; Kerala (Engg.) 2002]
  A)
           \[\mathbf{u}=0\] done clear
  B)
           \[\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}\] done clear
  C)
           \[\mathbf{u}=2\mathbf{a}\] done clear
  D)
           \[\mathbf{u}=\mathbf{a}\] done clear
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• question_answer3)
  If \[\mathbf{a}=\mathbf{i}+2\mathbf{j}-2\mathbf{k},\,\,\mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}\]and \[\mathbf{c}=\mathbf{i}+3\mathbf{j}-\mathbf{k},\] then \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is equal to                                        [RPET 1989]
  A)
           \[20\mathbf{i}-3\mathbf{j}+7\mathbf{k}\] done clear
  B)
           \[20\mathbf{i}-3\mathbf{j}-7\mathbf{k}\] done clear
  C)
                \[20\mathbf{i}+3\mathbf{j}-7\mathbf{k}\] done clear
  D)
                None of these done clear
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• question_answer4)
  If \[\mathbf{\alpha }=2\mathbf{i}+3\mathbf{j}-\mathbf{k},\,\,\mathbf{\beta }=-\mathbf{i}+2\mathbf{j}-4\mathbf{k}\] and \[\mathbf{\gamma }=\mathbf{i}+\mathbf{j}+\mathbf{k},\] then \[(\mathbf{\alpha }\times \mathbf{\beta }).(\mathbf{\alpha }\times \mathbf{\gamma })\] is equal to     [MNR 1984; UPSEAT 2000; Orissa JEE 2005]
  A)
           60 done clear
  B)
           64 done clear
  C)
           74 done clear
  D)
           ? 74 done clear
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• question_answer5)
  If \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\mathbf{0},\] then                             [RPET 1995]
  A)
           \[|\mathbf{a}|\,=\,|\mathbf{b}|\,.\,|\mathbf{c}|\,=1\] done clear
  B)
           \[\mathbf{b}||\mathbf{c}\] done clear
  C)
           \[\mathbf{a}||\mathbf{b}\] done clear
  D)
           \[\mathbf{b}\,\bot \,\mathbf{c}\] done clear
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• question_answer6)
  \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is equal to [RPET 1995; Kurukshetra CEE 1998; MP PET 2003]
  A)
           \[(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}-(\mathbf{a}\,.\,\mathbf{a})\,\mathbf{b}\] done clear
  B)
           \[(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{a}-(\mathbf{b}\,.\,\mathbf{c})\,\mathbf{a}\] done clear
  C)
           \[(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}\] done clear
  D)
           \[(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}-(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}\] done clear
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• question_answer7)
  If \[\mathbf{a}\times \mathbf{b}=\mathbf{c},\,\,\mathbf{b}\times \mathbf{c}=\mathbf{a}\] and a, b, c be moduli of the vectors a, b, c respectively, then
  A)
           \[a=1,\,\,b=c\] done clear
  B)
           \[c=1,\,\,a=1\] done clear
  C)
           \[b=2,\,\,c=2a\] done clear
  D)
           \[b=1,\,\,c=a\] done clear
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• question_answer8)
  If \[\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k},\] \[\mathbf{b}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}-2\mathbf{j}+2\mathbf{k},\]then \[(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] is equal to         
  A)
           \[24\mathbf{i}+7\mathbf{j}-5\mathbf{k}\] done clear
  B)
           \[7\mathbf{i}-24\mathbf{j}+5\mathbf{k}\] done clear
  C)
           \[12\mathbf{i}+3\mathbf{j}-5\mathbf{k}\] done clear
  D)
           \[\mathbf{i}+\mathbf{j}-7\mathbf{k}\] done clear
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• question_answer9)
  \[\mathbf{i}\times (\mathbf{j}\times \mathbf{k})=\] [RPET 1988; MP PET 1997]
  A)
           1 done clear
  B)
           0 done clear
  C)
           ? 1 done clear
  D)
           None of these done clear
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• question_answer10)
  If three unit vectors a, b, c are such that \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}}{2},\] then the vector a makes with b and c respectively the angles                                                [MP PET 1998]
  A)
           \[{{40}^{o}},\,{{80}^{o}}\] done clear
  B)
           \[{{45}^{o}},\,\,{{45}^{o}}\] done clear
  C)
           \[{{30}^{o}},\,\,{{60}^{o}}\] done clear
  D)
           \[{{90}^{o}},\,\,{{60}^{o}}\] done clear
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• question_answer11)
  Let \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}+\mathbf{j}.\] If c is a vector such that \[\mathbf{a}\,.\,\mathbf{c}=\,|\mathbf{c}|,\,\,|\mathbf{c}-\mathbf{a}|\,=2\sqrt{2}\]and the angle between \[(\mathbf{a}\times \mathbf{b})\] and c is \[{{30}^{o}}\], then \[|\,(\mathbf{a}\times \mathbf{b})\times \mathbf{c}|\,=\]                                       [IIT 1999]
  A)
           \[\frac{2}{3}\] done clear
  B)
           \[\frac{3}{2}\] done clear
  C)
           2 done clear
  D)
           3 done clear
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• question_answer12)
  \[i\times (j\times k)+j\times (k\times i)\,+k\times (i\times j)\] equals    [RPET 1999]
  A)
           i done clear
  B)
           j done clear
  C)
           k done clear
  D)
           0 done clear
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• question_answer13)
  \[[\mathbf{b}\times \mathbf{c}\,\,\mathbf{c}\times \mathbf{a}\,\,\mathbf{a}\times \mathbf{b}]\] is equal to [MP PET 2004]
  A)
           \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] done clear
  B)
           \[2\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done clear
  C)
           \[{{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}^{2}}\] done clear
  D)
           \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done clear
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• question_answer14)
  Given three unit vector \[a,\,b,\,c\] such that \[\mathbf{a}\,\bot \,\mathbf{b}\] and \[a||c\], then \[a\times (b\times c)\] is                                      [AMU 1999]
  A)
           a done clear
  B)
           b done clear
  C)
           c done clear
  D)
           0    done clear
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• question_answer15)
  If \[\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\,\mathbf{b}=\mathbf{i}-\mathbf{j}+\mathbf{k},\]\[\mathbf{c}=\mathbf{i}-\mathbf{j}-\mathbf{k}\], then \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is                                                [MP PET 2000]
  A)
           i ? j + k done clear
  B)
           2i ? 2j  done clear
  C)
           3i ? j + k done clear
  D)
           2i + 2j ? k done clear
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• question_answer16)
  If \[\overrightarrow{A}=i-2j-3k,\,\overrightarrow{B}=2i+j-k,\,\overrightarrow{C}=i+3j-2k\], then \[(\overrightarrow{A}\times \overrightarrow{B})\times \overrightarrow{C}\] is [MP PET 2001]
  A)
           \[5(-\,i+3j+4k)\] done clear
  B)
           \[4(-\,i+3j+4k)\] done clear
  C)
           \[5(-\,i-3j-4k)\] done clear
  D)
           \[4(i+3j+4k)\] done clear
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• question_answer17)
  \[a\times (b\times c)+b\times (c\times a)+c\times (a\times b)=\] [RPET 2003]
  A)
           0 done clear
  B)
           2[a b c] done clear
  C)
           a + b + c done clear
  D)
           3[a b c] done clear
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• question_answer18)
  Let \[a,\,b,\,c\] be three vectors from \[a\times (b\times c)=(a\times b)\times c\], if                                                      [Orissa JEE 2003]
  A)
           \[b\times (a\times c)=0\] done clear
  B)
           \[a(b\times c)=0\] done clear
  C)
           \[c\times a=a\times b\] done clear
  D)
           \[c\times b=b\times a\] done clear
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• question_answer19)
  If \[a=i+j+k,\,\,b=i+j,\,\,c=i\] and \[(a\times b)\times c=\lambda \,a+\mu \,b\], then \[\lambda +\mu =\] [EAMCET 2003]
  A)
                 0 done clear
  B)
                1 done clear
  C)
                2 done clear
  D)
                3 done clear
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• question_answer20)
  If the vectors a and b are mutually perpendicular, then \[\mathbf{a}\times \{\mathbf{a}\times \{\mathbf{a}\times (\mathbf{a}\times \mathbf{b})\}\}\] is equal to
  A)
           \[|\mathbf{a}{{|}^{2}}\mathbf{b}\] done clear
  B)
           \[|\mathbf{a}{{|}^{3}}\mathbf{b}\] done clear
  C)
           \[|\mathbf{a}{{|}^{4}}\mathbf{b}\] done clear
  D)
           None of these done clear
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• question_answer21)
  If a, b, c, d are coplanar vectors, then \[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=\]                                                      [MP PET 1998]
  A)
           \[|\,\mathbf{a}\,\times \,\mathbf{c}{{|}^{2}}\] done clear
  B)
           \[|\mathbf{a}\times \mathbf{d}{{|}^{2}}\] done clear
  C)
           \[|\mathbf{b}\times \mathbf{c}{{|}^{2}}\] done clear
  D)
           0 done clear
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• question_answer22)
  \[a\times [a\times (a\times \mathbf{b})]\] is equal to                                       [AMU 2001]
  A)
           \[(a\times a\,)\,.\,(b\times a)\] done clear
  B)
           \[a\,.\,(b\times a)-b\,.\,(a\times b)\] done clear
  C)
           \[[a\,.\,(a\times b)]\,a\] done clear
  D)
           \[(a\,.\,a)\,(b\times a)\] done clear
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• question_answer23)
  Given the following simultaneous equations for vectors         x and y \[\mathbf{x}+\mathbf{y}=\mathbf{a}\]                    .....(i) \[\mathbf{x}\times \mathbf{y}=\mathbf{b}\]             .....(ii) \[\mathbf{x}\,.\,\mathbf{a}=1\]                                .....(iii) Then \[\mathbf{x}=.........,\,\,\,\mathbf{y}=.......\]                              [Roorkee 1994]
  A)
           \[\mathbf{a},\,\,\mathbf{a}-\mathbf{x}\] done clear
  B)
           \[\mathbf{a}-\mathbf{b},\,\,\mathbf{b}\] done clear
  C)
           \[\mathbf{b},\,\,\mathbf{a}-b\] done clear
  D)
           None of these done clear
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• question_answer24)
  \[(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})=\]                 [MP PET 1997]
  A)
           [b c a] a done clear
  B)
           [c a b] b done clear
  C)
           [a b c] c done clear
  D)
           [a c b] b done clear
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